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The Euler-Maruyama approximations for the CEV model
1. | School of Mathematical Sciences, Monash University, Clayton Campus, Building 28, Wellington road, Victoria, 3800, Australia |
2. | School of Mathematical Sciences, Monash University, Clayton Campus, Building 28,, Wellington road, Victoria, 3800, Australia |
3. | Department of Engineering Systems, Tel Aviv University, Tel Aviv, Ramat Aviv, 69978, Israel |
References:
[1] |
A. Alfonsi, Higher order discretization schemes for the CIR process: application to affine term structure and Heston models, Mathematics of Computation, 79 (2010), 209-237.
doi: 10.1090/S0025-5718-09-02252-2. |
[2] |
V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function, Probability Theory and Related Fields, 104 (1996), 43-60.
doi: 10.1007/BF01303802. |
[3] |
P. Billingsley, "Converges of Probability Measures," John Wiley & Sons, New York, 1968. |
[4] |
M. Bossy and A. Diop, An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form $|x|^\alpha, \alpha\in [1/2, 1)$, preprint, version 2, INRIA, France, 2007. |
[5] |
J. C. Cox, The constant elasticity of variance option. Pricing model, The Journal of Portfolio Management, 23 (1997), 15-17. |
[6] |
D. Dawson, http://www.math.ubc.ca/ db5d/SummerSchool09/lectures-dd/lecture4.pdf, (Accessed on August 3, 2010.) |
[7] |
G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term, Applied Stochastic Models and Data Analysis, 14 (1998), 77-84.
doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2. |
[8] |
F. Delbaen and H. A. Shirakawa, Note of option pricing for constant elasticity of variance model, Asia-Pacific Financial Markets, 9 (2002), 159-168.
doi: 10.1023/A:1024173029378. |
[9] |
W. Feller, Two singular diffusion problems, Annals of Mathematics, 54 (1951), 173-182.
doi: 10.2307/1969318. |
[10] |
I. Gy, A note on Eulers approximations, Potential Analysis, 8 (1998), 205-216.
doi: 10.1023/A:1008605221617. |
[11] |
I. Gy,I. ongy and N. Krylov, Existence of strong solutions for Itó's stochastic equations via approximations, Probability Theory and Related Fields, 105 (1996), 143-158.
doi: 10.1007/BF01203833. |
[12] |
N. Halidias and P. Kloeden, A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient, BIT Numerical Mathematics, 48 (2008), 51-59 .
doi: 10.1007/s10543-008-0164-1. |
[13] |
P. L. Hennequin and A. Tortrat, "Théorie des Probabilités et Quelques Applications," Masson et Cie, Éditeurs, Paris, 1965. |
[14] |
D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process, Computational Finance, 8 (2005), 35-61. |
[15] |
D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal of Numerical Analysis, 40 (2002), 1041-1063.
doi: 10.1137/S0036142901389530. |
[16] |
M. Hutzenthaler and A. Jentzen, Non-globally Lipschitz counterexamples for the stochastic Euler scheme, preprint, arXiv:0905.0273. |
[17] |
A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coeffcients, Numerische Mathematik, 112 (2009), 41-64.
doi: 10.1007/s00211-008-0200-8. |
[18] |
Yu. Kabanov, R. Liptser and A. N. Shiryaev, Estimates of closeness in variation of probability measures (Russian), Dokl. Akad. Nauk SSSR, 278 (1984), 265-268. |
[19] |
F. C. Klebaner, "Introduction to Stochastic Calculus with Applications," 2nd edition, Imperial College Press, London, 2005. |
[20] |
P. E. Kloeden and K. Platten, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag, Berlin, 1992. |
[21] |
R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales" [Translated from the Russian by K. Dzjaparidze] Mathematics and its Applications (Soviet Series), 49 Kluwer Academic Publishers Group, Dordrecht, 1989. |
[22] |
G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics," Springer-Verlag, Berlin, 2004. |
[23] |
L. C. G. Rogers and D. Williams, "Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus," Reprint of the 2nd edition, Cambridge University Press, Cambridge, 2000. |
[24] |
T. Shiga and S. Watanabe, Bessel diffusions as a one parameter family of diffusion processes, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 27 (1973), 37-46. |
[25] |
D. Talay, Simulation and numerical analysis of stochastic differential systems: A review, in "Probabilistic Methods in Applied Physics. Lecture Notes in Physics," 451, 63-106, Springer, Berlin, 1995. |
[26] |
L. Yan, The Euler scheme with irregular coefficients, Annals of Probability, 30 (2002), 1172-1194.
doi: 10.1214/aop/1029867124. |
[27] |
H. Zähle, Weak approximation of SDEs by discrete time processes, Journal of Applied Mathematics and Stochastic Analysis, 2008 (2008), Article ID 275747, 15 pages. |
show all references
References:
[1] |
A. Alfonsi, Higher order discretization schemes for the CIR process: application to affine term structure and Heston models, Mathematics of Computation, 79 (2010), 209-237.
doi: 10.1090/S0025-5718-09-02252-2. |
[2] |
V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function, Probability Theory and Related Fields, 104 (1996), 43-60.
doi: 10.1007/BF01303802. |
[3] |
P. Billingsley, "Converges of Probability Measures," John Wiley & Sons, New York, 1968. |
[4] |
M. Bossy and A. Diop, An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form $|x|^\alpha, \alpha\in [1/2, 1)$, preprint, version 2, INRIA, France, 2007. |
[5] |
J. C. Cox, The constant elasticity of variance option. Pricing model, The Journal of Portfolio Management, 23 (1997), 15-17. |
[6] |
D. Dawson, http://www.math.ubc.ca/ db5d/SummerSchool09/lectures-dd/lecture4.pdf, (Accessed on August 3, 2010.) |
[7] |
G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term, Applied Stochastic Models and Data Analysis, 14 (1998), 77-84.
doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2. |
[8] |
F. Delbaen and H. A. Shirakawa, Note of option pricing for constant elasticity of variance model, Asia-Pacific Financial Markets, 9 (2002), 159-168.
doi: 10.1023/A:1024173029378. |
[9] |
W. Feller, Two singular diffusion problems, Annals of Mathematics, 54 (1951), 173-182.
doi: 10.2307/1969318. |
[10] |
I. Gy, A note on Eulers approximations, Potential Analysis, 8 (1998), 205-216.
doi: 10.1023/A:1008605221617. |
[11] |
I. Gy,I. ongy and N. Krylov, Existence of strong solutions for Itó's stochastic equations via approximations, Probability Theory and Related Fields, 105 (1996), 143-158.
doi: 10.1007/BF01203833. |
[12] |
N. Halidias and P. Kloeden, A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient, BIT Numerical Mathematics, 48 (2008), 51-59 .
doi: 10.1007/s10543-008-0164-1. |
[13] |
P. L. Hennequin and A. Tortrat, "Théorie des Probabilités et Quelques Applications," Masson et Cie, Éditeurs, Paris, 1965. |
[14] |
D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process, Computational Finance, 8 (2005), 35-61. |
[15] |
D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal of Numerical Analysis, 40 (2002), 1041-1063.
doi: 10.1137/S0036142901389530. |
[16] |
M. Hutzenthaler and A. Jentzen, Non-globally Lipschitz counterexamples for the stochastic Euler scheme, preprint, arXiv:0905.0273. |
[17] |
A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coeffcients, Numerische Mathematik, 112 (2009), 41-64.
doi: 10.1007/s00211-008-0200-8. |
[18] |
Yu. Kabanov, R. Liptser and A. N. Shiryaev, Estimates of closeness in variation of probability measures (Russian), Dokl. Akad. Nauk SSSR, 278 (1984), 265-268. |
[19] |
F. C. Klebaner, "Introduction to Stochastic Calculus with Applications," 2nd edition, Imperial College Press, London, 2005. |
[20] |
P. E. Kloeden and K. Platten, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag, Berlin, 1992. |
[21] |
R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales" [Translated from the Russian by K. Dzjaparidze] Mathematics and its Applications (Soviet Series), 49 Kluwer Academic Publishers Group, Dordrecht, 1989. |
[22] |
G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics," Springer-Verlag, Berlin, 2004. |
[23] |
L. C. G. Rogers and D. Williams, "Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus," Reprint of the 2nd edition, Cambridge University Press, Cambridge, 2000. |
[24] |
T. Shiga and S. Watanabe, Bessel diffusions as a one parameter family of diffusion processes, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 27 (1973), 37-46. |
[25] |
D. Talay, Simulation and numerical analysis of stochastic differential systems: A review, in "Probabilistic Methods in Applied Physics. Lecture Notes in Physics," 451, 63-106, Springer, Berlin, 1995. |
[26] |
L. Yan, The Euler scheme with irregular coefficients, Annals of Probability, 30 (2002), 1172-1194.
doi: 10.1214/aop/1029867124. |
[27] |
H. Zähle, Weak approximation of SDEs by discrete time processes, Journal of Applied Mathematics and Stochastic Analysis, 2008 (2008), Article ID 275747, 15 pages. |
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